Transformations

Cat Rotating Around a Bouncing Ball

/ ivasallay / Leave a comment

Not every topic in my mathematics education was covered equally well. I don’t recall learning anything about geometric transformations when I was in school. Many years after I graduated from college, when I was teaching mathematics, I learned how to make a shape rotate around the origin:

For each original point (x, y),
(-y, x) maps it 90° counterclockwise around the origin,
(y, -x) maps it 90° clockwise around the origin, and
(-x, -y) maps it 180° around the origin.

This year, the 9th-graders I work with at school need to know how to rotate a shape around a point that is NOT the origin. This is a topic I had never thought about before. To patch up this hole in my math knowledge, I decided to play with rotations in Desmos.

I began with
the point (-1, 2) and
the endpoints (2, 3) and (4, 7), and
the polygon function in Desmos to connect the endpoints.

I noticed that the point and the line segment had the same relationship as
(0, 0) and
the line segment with endpoints (3, 1) and (5, 5).
I noted that the following coordinates worked beautifully to rotate the line segment:
(x-1, y+2) maps it onto the line segment’s original endpoints, (2, 3) and (4, 7).
(-y-1, x+2) maps it 90° counterclockwise around (-1, 2),
(y-1, -x+2) maps it 90° clockwise around (-1, 2), and
(-x-1, -y+2) maps it 180° around (-1, 2).

I was quite pleased with the symmetry of those relationships, so I decided to add a few more ordered pairs to my table in Desmos: (4,-3), (6, 3), and (-2, 4). Desmos’s polygon function automatically rotated the line segments produced by those ordered pairs, and this lovely symmetrical design was produced:

Next, I wondered what would happen if I changed the center of rotation. I made an ordered pair, (a, b) and used sliders to move the point around. Now the point reminded me of a ball, and I got the idea to make the rotated shape look like a cat playing with the ball. I made the original cat a little darker than the rotated ones. This was the result:

That cat took a lot of ordered pairs to make. I got to thinking about how rotations are often used in tessellations. I found a simpler-looking kitten that tessellates and recreated it in this Desmos graph. Again, the original cat is darker than the rotated ones.

No cats were harmed in the production of these graphs.

Which of the cats do you like the best, or would you have used a different animal to play with the ball?

1743 Finding Ways to Transform My Heart

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Today’s Puzzle:

An equation of a unit circle centered at the origin is x² + y² = 1.

If we change just the “y” part of that equation, we can get a lovely heart just in time for Valentine’s Day. Try it yourself by typing the equations into Desmos.

There are other mathematical equations for a heart, but this is the one I’m exploring in this post.

I was puzzled over how I could transform that heart. Can I make it bigger, or dilate it? Can I slide it away from the origin or translate it? Can I rotate it? Can I reflect it across the x or y-axis?

These are questions I’d like you to explore as well.

Heart Dilation:

In this first graphic, I was able to make my heart bigger. What kind of math let me do that? Also, how did I color the inside of some of the hearts? Look at the equations next to the heart and try to figure it out. The concentric hearts are evenly spaced. Do you recognize a pattern in the numbers that made that happen?

Heart Slide (Translation):

If we changed the center of a circle to (a, -b) instead of the origin, we would slide the whole circle. Here’s how we change the equation of the circle to give it a new center:
(x-a)²+(y+b)² = 1.

Similarly, in the next graphic, I was able to slide my heart away from the origin. How did I do that? Look at the equations to see how.

Heart Rotation:

A circle looks the same no matter how it is rotated, but the same isn’t true for a heart. Look at the equations below. How was I able to rotate my heart around the origin?

Heart Reflection:

Since a heart is symmetric, its reflection across the x-axis doesn’t look that interesting to me. Instead, I created a double heart that I reflected across both the x-axis and the y-axis:

Just for Fun:

Next, I was curious about what would happen if I changed the exponents on the outside of the parenthesis, so I changed a 2 from my original equation to an 8 in a couple of different places as I moved the heart from left to right. How did changing the exponent affect my heart? I found that as long as the exponent stays even, it still looks a little like a heart.

I was also curious about what would happen to my heart if I changed the “2/3” to a different fraction. I used fractions less than one as well as fractions greater than one. For many of my fractions, I used the post number, 1743, as the denominator. As long as the numerator was even and the denominator was odd, the graph still looked mostly like a heart. However, the closer the fraction was to zero, the more it looked like a circle.

Finally, I created this lovely flower using some of what I learned by making these transformations:

And for just a little bit more fun, I created a simple but chaotic-looking animation that I’ve titled Hearts in Motion. Enjoy!

I had so much fun exploring this heart in Desmos. Thank you for allowing me to share my excitement with you. Here’s a different Desmos heart created by a reader and shared with me. Click on it to make the heart beat:

 

Factors of 1743:

  • 1743 is a composite number.
  • Prime factorization: 1743 = 3 × 7 × 83.
  • 1743 has no exponents greater than 1 in its prime factorization, so √1743 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1743 has exactly 8 factors.
  • The factors of 1743 are outlined with their factor pair partners in the graphic below.

More About the Number 1743:

1743 is the difference of two squares in four different ways:
872² – 871² = 1743,
292² – 289² = 1743,
128² – 121² = 1743, and
52² – 31² = 1743.